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Notes -
A Big Day in the Culture War
Apologies for any incoherency or grammatical errors; today's events led me to down more than my usual share of booze.
Earlier today, big things happened in math. No, not Claude providing a sub quadratic 3SUM. OpenAI released hundreds of notable math results, in a GitHub repo.
Fun results:
Hilbert's tenth problem over (\mathbb{Q}) is undecidable
The quasi-Riemann hypothesis
The rational Hodge conjecture holds for every CM abelian variety
Integer multiplication can be done in sub-log-linear time. Oh, there's also a sub-log linear DFT.
Thoughts and observations:
Mathematicians are big mad. View the relevant subreddit on our progenitor site. Most there are probably, at best, adjuncts at community colleges desperately coping with the downward trajectory of already marginal careers, but it's fair to say that the writing is on the wall for mathematicians. There's probably a double digit number of grad students staring into a glass of whiskey tonight and thinking of hanging themselves.
My immediate question was about whether this closer to the current peak of performance, or just a lazy demonstration of OpenAI's power. So, I took one of the particularly interesting preprints to me (memory and precision in Gaussian models) and tested whether it's at the edge of capabilities or not. 30 minutes of back and forth with Astra (itself behind OAI's internal model) resulted in a significantly stronger result, on multiple dimensions. While I finished my first bottle, I spent a fair amount of time convincing myself of the result; I was convinced the strengthened results were plausible. Write this off as AI psychosis if you want, but try it yourself; I'm genuinely curious for what you get. (The back and forth, here, was entirely me saying "you can do it!", "keep at it, I believe in you", and "you've got this, finish it!")
Probably the most important line in OAI's announcement post is "The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking." This isn't a case of OAI spending millions for a marketing bump. My bet is a kind of Pareto distribution: most took minutes, not hours, with a long tail around Riemann-level results pulling up the average significantly.
Notably, ML related results are nearly entirely absent from this batch of proofs; maybe a half dozen touch on it, distantly. Some problem indices are skipped in overview.md. Conspiratorially, my inclination was to think they filtered them out for competitive advantage. I can't find any evidence of that in the GitHub repo or any of the preprints (equally plausible: deduping), so maybe they judiciously decided not to point their mathematical ballista at ML. ML also doesn't have a meaningful bank of rigorous conjectures, so given the conjecture sources, maybe they're not yet digging into ML math. But color me skeptical.
Does math matter? Is it something to advance civilization and technology, or an artistic pasttime for humans to create logical beauty? Likely both, today, but this is an almost nuclear detonation against the latter.
My
understandingimpression of the navier-stokes "result" is that it is a mound of Lean and a paper no person can parse. Lean kernels say proof is valid, but Lean kernels may be buggy. The problem itself is so convoluted there is no agreement if even the formulation of the problem itself is correct, so Lean not being buggy is not enough either. Sounds unreliable, I don't care and presume the result is wrong.In near-ish future, human understanding not needed, I'll take whatever pure math result AI publishes on faith alone. For now, show me something genuinely brilliant that any competent mathematican can verify in under a week (any of that in this pile? probably, but let's see in a month or 3), or a practical, usable result.
Break some encryption scheme, show a big advancement in ML, right, you totally can (true) but won't release. Any big applied math result, is that permitted, can american AI corpos into public science beyond IPO pump curios?
Who doubts this? The formulation has been part of mathlib for some time already AFAIK.
Dumb of me to phrase it that way, entire sentence is confused. I meant to say, correct in the sense of formulating the core problem the big deal is about, not just a problem; enough, to count towards the core problem - no bearing on correct. Who doubts, I'd pretend to know if I listed anything concrete, it only is an impression from some scattered reading.
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My heuristic to such results is that a counterexample is easier to verify then a positive proof. With Navier-Stokes conjecture in particular, the fact that another team was close to publishing a (different) counterexample makes me almost certain that Navier-Stokes conjecture doesn't hold. So if I were working on something that relied on the conjecture to be true in all cases, I would certainly redirect my efforts. (Or, as is quite common in research of pure mathematics, restrict my results to the conditions where the N-S conjecture holds true.)
Can you explain to a guy that's not deep into the math trenches what it would mean for that conjecture to be false in concrete terms irl? Or is that part of those abstractions on abstractions the mathematicians sometimes get themselves into?
In concrete terms the direct meaning of a blowup in the incompressible Navier-Stokes equations is nothing - incompressible fluids do not exist. For that matter, fluids do not exist - it's all atoms, and we pretend that atoms blend into a perfect continuum because it's a lot easier to approximate a problem using a million finite elements rather than a septillion atoms. Perhaps most fluid flow simulations are performed using incompressible Navier-Stokes anyway, though. If some of the fluid gets significantly warmer than the rest we'll partially take compressibility into account by adding a Boussinesq buoyancy term, but otherwise we generally pretend a fluid is incompressible until we're dealing with speeds near the speed of sound (in that fluid - e.g. "Mach 1" might mean 767 mph in standard air, but Mach 1 is 4 times faster in water), and we pretend flowing matter is a fluid until we're dealing with such tiny length scales or rarified gases that you have to worry about atoms slipping right past each other.
IMHO the rest of what's interesting is "abstractions on abstractions", especially for now, but ones that might lead to practical consequences in the future.
Half of what's interesting in the most practical sense here is: we want to control the error in our fluid flow simulations, and this might be another step toward figuring out how to do so more rigorously.
That "million finite elements" (in practice more likely finite volumes; unimportant distinction here) doesn't give you an exact solution to your flow problem, just an approximate solution, and that's fine because it's engineers who need the approximate solution and engineers use safety factors and the safety factors mean that if we give them a solution that's 10% off they're still fine. I once found a bug I'd written that made my code 0.0001% off, which snowballed through other equations to make a ton of engineering solutions 0.1% off. Coming from academia I was mortified, but the engineers hadn't noticed anything was wrong until I fixed the bug and thereby tripped some regression tests against past "gold" solutions. They hadn't yet bothered setting up problems where they could compare simulation solutions to exact solutions to verify the error magnitudes. Their idea of responsible testing was just validating against experiments, and the experimental measurement error was well over 0.1% so the validations said "this matches experiments well enough", and thankfully we never were simulating anything where simulation and reality diverged more than they did in experiments.
The trouble is that, even with an approximate solution, we always like to know how approximate it is, and if we always had an experiment to validate against then we wouldn't bother running a simulation. Grossly oversimplifying: with the simplest partial differential equations, we can prove results that look like "if the boundary conditions and 'forcing function' are of magnitude ‖f‖, then the solution u must be of magnitude ‖u‖≤c‖f‖, for a constant c that depends on the boundary". We can't actually find the solution u exactly, but we can approximate it, and we can turn those results about the exact solution into results about the error e=u-uₕ on an approximate solution uₕ, results that look something like ‖e‖≤mc‖f‖, for a constant m that depends on what type of approximation we use and how expensive we make it. If we want to guarantee a small error, for a given c and ‖f‖ we can pick an approximation with m small enough to get it.
We couldn't do this with Navier-Stokes in the most general cases, and now it's clear why: this does not work if c is infinity. Techniques that use the fact that the solution can't grow too badly to prove that our error can't grow too badly do not work if the solution can blow up. Often we can still prove that we're dealing with a "nice" case where the fluid viscosity can "overpower" a weak forcing enough to prevent a blowup, but now we know that "often" isn't "always"; we weren't just missing some clever trick to get there. This counterexample might stop people from wandering down a ton of blind alleys looking for one, or by showing us what we need to avoid it might point the way to expanding our criteria for "nice".
And the other half of what might become interesting down the road here: We don't understand turbulence nearly well enough, and yet most of the liquid flow and essentially all of the gas flow problems we want to solve are turbulent.
If we have a turbulent flow, that grid of "a million finite elements" becomes laughably small for a direct numerical simulation of it, because smaller and smaller turbulent eddies quickly become so small that we can't represent them accurately, while still being powerful enough to greatly affect the flow on the larger "macroscale" we care about. Instead we need to use, say, "35 trillion grid points", if we want a solution to just the equations of fluid flow that resembles the actual fluid flow. We can't afford to make every engineering problem millions of times more expensive to solve. So we've come up with "turbulence models", additional equations that try to approximate the statistically-averaged-out effects of turbulence on the macroscale flow, rather than precisely tracking every microscale vortex from its creation until viscosity damps it out. Now our equations are only 10% more expensive to solve (with the cheapest models) or 50% or 200% or whatever as we switch to better and better models, and we hope the results are good enough. It really is a mix of "hope" and "this matches experiments well enough" (for a much looser definition of "well enough" this time), because we don't have any really good turbulence models, not in the same "so close we never notice a difference" or "this is an inevitable consequence of the laws of physics with a few simplifications" senses in which incompressible-Navier-Stokes is a good model for viscous macroscopic fluid flow. Heisenberg supposedly said "When I meet God, I am going to ask him two questions: why relativity? And why turbulence? I really believe he will have an answer for the first." - apocryphal quote from nearly a century ago, but the turbulence experts I've talked to during the 21st century didn't disagree. But now we've got some fairly unexpected Navier-Stokes results, specifically dealing with the ways in which viscosity can or can't damp down a microscale vortex? This might go nowhere, or it might end up leading someone to a much better turbulence model down the road. I haven't looked at this stuff in a decade, but at a brief glance right now it looks like the cutting-edge in turbulence modeling is split between "tweak some 1990-era models to make them a bit better" and "if we enforce a few laws of physics on top of a machine-learning black-box it sort of works". Breaking out of those ruts wouldn't have a flashy Millennium Prize in the headlines but it could be a huge breakthrough.
That was super-informative! I just gained more useful insight from 2 minutes of reading your comment then I did from an hour of reading other explainers of the Navier-Stokes result. Many thanks!
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